Given a prototype digital
filter design, transformations similar to the bilinear transform
can also be developed.
Requirements on such a mapping
z1=gz1
z
1
g
z
1
:
 points inside the unit circle stay inside the unit
circle (condition to preserve stability)
 unit circle is mapped to itself (preserves frequency
response)
This condition implies
that
e−(i
ω
1
)=ge−(iω)=gωei∠gω
ω
1
g
ω
g
ω
g
ω
requires that
ge−(iω)=1
g
ω
1
on the unit circle!
Thus we require an allpass
transformation:
gz1=∏
k
=1pz1−
α
k
1−
α
k
z1
g
z
1
k
1
p
z
1
α
k
1
α
k
z
1
where

α
K
<1
α
K
1
, which is required to satisfy this condition.
z
1
1=z1−a1−az1
z
1
1
z
1
a
1
a
z
1
which maps original filter with a cutoff at
ωc
ωc
to a new filter with cutoff
ωc
′
ωc
′
,
a=sin12(
ω
c
−
ω
c
′
)sin12(
ω
c
+
ω
c
′
)
a
1
2
ω
c
ω
c
′
1
2
ω
c
ω
c
′
z
1
1=z1+a1+az1
z
1
1
z
1
a
1
a
z
1
which maps original filter with a cutoff at
ωc
ωc
to a frequency reversed filter with cutoff
ωc
′
ωc
′
,
a=cos12(
ω
c
−
ω
c
′
)cos12(
ω
c
+
ω
c
′
)
a
1
2
ω
c
ω
c
′
1
2
ω
c
ω
c
′
(Interesting and occasionally useful!)